Write any five rational numbers between - 14 15 and 1 5 .

Given rational numbers are - 14 15 and 1 5 . First, let us convert - 14 15 and 1 5 into rational numbers with the same denominators by taking LCM of denominators. LCM of 15 and 5 is 15 We have, 1 5 = 1 × 3 5 × 3 = 3 15 (multiply numerator and denominator by 3 ) Therefore, five rational numbers between - 14 15 = - 14 15 and 1 5 = 3 15 are - 13 15 , - 12 15 , - 11 15 , - 10 15 , - 9 15 Hence, five rational numbers between - 14 15 and 1 5 are - 13 15 , - 12 15 , - 11 15 , - 10 15 , - 9 15 .

Verify the property, x × ( y + z ) = x × y + x × z of rational numbers by taking x = - 1 2 , y = 3 4 , z = 1 4 .

We need to verify, - 1 2 3 4 + 1 4 = - 1 2 × 3 4 + - 1 2 × 1 4 . We have, For all rational numbers x , y and z , x ( y + z ) = x y + x z . . . . . ( i ) Let the three rational numbers be x = - 1 2 , y = 3 4 , z = 1 4 Let us substitute the values in LHS of i , we get x ( y + z ) = - 1 2 3 4 + 1 4 = - 1 2 3 + 1 4 = - 1 2 4 4 = - 1 2 . . . . . . . . 1 Now, let us check RHS of i , x × y + x × z = - 1 2 × 3 4 + - 1 2 × 1 4 = - 3 8 + - 1 8 = - 3 - 1 8 = - 4 8 = - 1 2 . . . . . . . . 2 Therefore, from ( 1 ) and ( 2 ) , we get: - 1 2 3 4 + 1 4 = - 1 2 × 3 4 + - 1 2 × 1 4 Hence, we conclude that x × ( y + z ) = x × y + x × z .

Verify the property x × ( y + z ) = x × y + x × z of rational numbers by taking x = - 1 2 , y = 2 3 , z = 3 4 .

We need to verify, - 1 2 2 3 + 3 4 = - 1 2 × 2 3 + - 1 2 × 3 4 . We have, For all rational numbers x , y and z , x ( y + z ) = x y + x z . . . . ( 1 ) The given values of three rational numbers are x = - 1 2 , y = 2 3 , z = 3 4 Substitute the values in LHS of 1 , we get: x ( y + z ) = - 1 2 2 3 + 3 4 = - 1 2 2 × 4 + 3 × 3 12 = - 1 2 8 + 9 12 = - 1 2 17 12 = - 17 24 . . . . . . . . i Now, let us check RHS of 1 , we get: x × y + x × z = - 1 2 × 2 3 + - 1 2 × 3 4 = - 2 6 + - 3 8 = - 8 - 9 24 = - 17 24 . . . . . . . . ii Therefore, from i and ii , we get: - 1 2 2 3 + 3 4 = - 1 2 × 2 3 + - 1 2 × 3 4 Hence, we conclude that x × ( y + z ) = x × y + x × z .